Injective metrics on buildings and symmetric spaces

TL;DR

This paper explores injective metrics on symmetric spaces and buildings, showing their properties and actions of classical groups, with implications for geometric group theory and metric space embeddings.

Contribution

It introduces natural injective and coarsely injective metrics on classical symmetric spaces and buildings, and proves proper cocompact actions of classical groups on these spaces.

Findings

Goldman-Iwahori metric satisfies Helly property for balls

Classical semisimple groups act properly and cocompactly on injective spaces

Injective hull of GL(n,R) is the space of all norms on R^n

Abstract

In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a natural piecewise $\ell^\infty$ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural invariant Finsler metric, restricting to an $\ell^\infty$ on each flat, which is coarsely injective. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. We identify the injective hull of the symmetric space of $\operatorname{GL}(n,\mathbb{R})$ as the space of all norms on $\mathbb{R}^n$. The only exception is the special linear group: if $n=3$ or $n \geq 5$ and $\mathbb{K}$ is a local field, we show that $\operatorname{SL}(n,\mathbb{K})$ does not act properly and coboundedly on an injective metric space.